rosanna keefe - library of congress

Theories of Vagueness

Rosanna KeefeUniversity of Shef®eld

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# Rosanna Keefe 2000

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written permission of Cambridge University Press.

First published 2000

Printed in the United Kingdom at the University Press, Cambridge

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A catalogue record for this book is available from the British Library

Library of Congress Cataloguing in Publication data

Keefe, Rosanna.Theories of vagueness / Rosanna Keefe.

p. cm.Includes bibliographical references and index.

ISBN 0 521 65067 4 (hardback)1. Vagueness (Philosophy) I. Title.

B105.V33 K44 2000160±dc21

ISBN 0 521 65067 4 hardback

Contents

Acknowledgements page xi

Introduction 1

1 The phenomena of vagueness 6

2 How to theorise about vagueness 37

3 The epistemic view of vagueness 62

4 Between truth and falsity: many-valued logics 85

5 Vagueness by numbers 125

6 The pragmatic account of vagueness 139

7 Supervaluationism 152

8 Truth is super-truth 202

References 221Index 229

ix

1

The phenomena of vagueness

1. central features of vague expressions

The parties to the vigorous debates about vagueness largely agreeabout which predicates are vague: paradigm cases include `tall', `red',`bald', `heap', `tadpole' and `child'. Such predicates share threeinterrelated features that intuitively are closely bound up with theirvagueness: they admit borderline cases, they lack (or at least appar-ently lack) sharp boundaries and they are susceptible to soritesparadoxes. I begin by describing these characteristics.Borderline cases are cases where it is unclear whether or not the

predicate applies. Some people are borderline tall: not clearly tall andnot clearly not tall. Certain reddish-orange patches are borderline red.And during a creature's transition from tadpole to frog, there will bestages at which it is a borderline case of a tadpole. To offer at thisstage a more informative characterisation of borderline cases and theunclarity involved would sacri®ce neutrality between various com-peting theories of vagueness. Nonetheless, when Tek is borderlinetall, it does seem that the unclarity about whether he is tall is notmerely epistemic (i.e. such that there is a fact of the matter, we justdo not know it). For a start, no amount of further information abouthis exact height (and the heights of others) could help us decidewhether he is tall. More controversially, it seems that there is no factof the matter here about which we are ignorant: rather, it isindeterminate whether Tek is tall. And this indeterminacy is oftenthought to amount to the sentence `Tek is tall' being neither true norfalse, which violates the classical principle of bivalence. The law ofexcluded middle may also come into question when we considerinstances such as `either Tek is tall or he is not tall'.Second, vague predicates apparently lack well-de®ned extensions.

On a scale of heights there appears to be no sharp boundary between

6

the tall people and the rest, nor is there an exact point at which ourgrowing creature ceases to be a tadpole. More generally, if weimagine possible candidates for satisfying some vague F to be arrangedwith spatial closeness re¯ecting similarity, no sharp line can be drawnround the cases to which F applies. Instead, vague predicates arenaturally described as having fuzzy, or blurred, boundaries. Butaccording to classical logic and semantics all predicates have well-de®ned extensions: they cannot have fuzzy boundaries. So again thissuggests that a departure from the classical conception is needed toaccommodate vagueness.Clearly, having fuzzy boundaries is closely related to having

borderline cases. More speci®cally, it is the possibility of borderlinecases that counts for vagueness and fuzzy boundaries, for if all actuallyborderline tall people were destroyed, `tall' would still lack sharpboundaries. It might be argued that for there to be no sharp boundarybetween the Fs and the not-Fs just is for there to be a region ofpossible borderline cases of F (sometimes known as the penumbra).On the other hand, if the range of possible borderline cases betweenthe Fs and the not-Fs was itself sharply bounded, then F would havea sharp boundary too, albeit one which was shared with the border-line Fs, not with the things that were de®nitely not F. The thoughtthat our vague predicates are not in fact like this ± their borderlinecases are not sharply bounded ± is closely bound up with the key issueof higher-order vagueness, which will be discussed in more detail in§6.Third, typically vague predicates are susceptible to sorites para-

doxes. Intuitively, a hundredth of an inch cannot make a differenceto whether or not a man counts as tall ± such tiny variations,undetectable using the naked eye and everyday measuring instru-ments, are just too small to matter. This seems part of what it is for`tall' to be a vague height term lacking sharp boundaries. So we havethe principle [S1] if x is tall, and y is only a hundredth of an inchshorter than x, then y is also tall. But imagine a line of men, startingwith someone seven feet tall, and each of the rest a hundredth of aninch shorter than the man in front of him. Repeated applications of[S1] as we move down the line imply that each man we encounter istall, however far we continue. And this yields a conclusion which isclearly false, namely that a man less than ®ve feet tall, reached afterthree thousand steps along the line, is also tall.


7

Similarly there is the ancient example of the heap (Greek soros,from which the paradox derives its name). Plausibly, [S2] if x is a heapof sand, then the result y of removing one grain will still be a heap ±recognising the vagueness of `heap' seems to commit us to thisprinciple. So take a heap and remove grains one by one; repeatedapplications of [S2] imply absurdly that the solitary last grain is a heap.The paradox is supposedly owed to Eubulides, to whom the liarparadox is also attributed. (See Barnes 1982 and Burnyeat 1982 fordetailed discussion of the role of the paradox in the ancient world.)Arguments with a sorites structure are not mere curiosities: they

feature, for example, in some familiar ethical `slippery slope' argu-ments (see e.g. Walton 1992 and Williams 1995). Consider theprinciple [S3] if it is wrong to kill something at time t after concep-tion, then it would be wrong to kill it at time t minus one second.And suppose we agree that it is wrong to kill a baby nine monthsafter conception. Repeated applications of [S3] would lead to theconclusion that abortion even immediately after conception wouldbe wrong. The need to assess this kind of practical argumentationincreases the urgency of examining reasoning with vague predicates.Wright (1975, p. 333) coined the phrase tolerant to describe

predicates for which there is `a notion of degree of change too smallto make any difference' to their applicability. Take `[is] tall' (forsimplicity, in mentioning predicates I shall continue, in general, toomit the copula). This predicate will count as tolerant if, as [S1]claims, a change of one hundredth of an inch never affects itsapplicability. A tolerant predicate must lack sharp boundaries; for if Fhas sharp boundaries, then a boundary-crossing change, howeversmall, will always make a difference to whether F applies.1 Moreover,a statement of the tolerance of F can characteristically serve as theinductive premise of a sorites paradox for F (as in the example of `tall'again).Russell provides one kind of argument that predicates of a given

class are tolerant: if the application of a word (a colour predicate, forexample) is paradigmatically based on unaided sense perception, itsurely cannot be applicable to only one of an indiscriminable pair(1923, p. 87). So such `observational' predicates will be tolerant with

1 Note that throughout this book, when there is no potential for confusion I am casualabout omitting quotation marks when natural language expressions are not involved,e.g. when talking about the predicate F or the sentence p & :p.

Theories of vagueness

8

respect to changes too small for us to detect. And Wright develops, indetail, arguments supporting the thesis that many of our predicates aretolerant (1975 and 1976). In particular, consideration of the role ofostension and memory in mastering the use of such predicates appearsto undermine the idea that they have sharp boundaries which couldnot be shown by the teacher or remembered by the learner.Arguments of this kind are widely regarded as persuasive: I shall referto them as `typical arguments for tolerance'. A theory of vaguenessmust address these arguments and establish what, if anything, theysucceed in showing, and in particular whether they show that theinductive premise of the sorites paradox holds.Considerations like Russell's and Wright's help explain why vague

predicates are so common (whatever we say about the soritespremise). And they also seem to suggest that we could not operatewith a language free of vagueness. They make it dif®cult to seevagueness as a merely optional or eliminable feature of language. Thiscontrasts with the view of vagueness as a defect of natural languagesfound in Frege (1903, §56) and perhaps in Russell's uncharitablesuggestion (1923, p. 84) that language is vague because our ancestorswere lazy. A belief that vagueness is inessential and therefore unim-portant may comfort those who ignore the phenomenon. But theircomplacency is unjusti®ed. Even if we could reduce the vagueness inour language (as science is often described as striving to do byproducing sharper de®nitions, and as legal processes can accomplishvia appeal to precedents), our efforts could not in practice eliminate itentirely. (Russell himself stresses the persistent vagueness in scienti®cterms, p. 86; and it is clear that the legal process could never reachabsolute precision either.) Moreover, in natural language vaguepredicates are ubiquitous, and this alone motivates study of thephenomenon irrespective of whether there could be usable languagesentirely free of vagueness. Even if `heap' could be replaced by someterm `heap*' with perfectly sharp boundaries and for which no soritesparadox would arise, the paradox facing our actual vague term wouldremain.2 And everyday reasoning takes place in vague language, so noaccount of good ordinary reasoning can ignore vagueness.

2 See Carnap 1950, chapter 1, Haack 1974, chapter 6 and Quine 1981 on the replace-ment of vague expressions by precise ones, and see Grim 1982 for some dif®cultiesfacing the idea. Certain predicates frequently prompt the response that there is in fact asharp boundary for their strict application, though we use them more loosely ± in par-


9

In the next section I shall discuss the variety of vague expressions ±a variety which is not brought out by the general form of argumentsfor tolerance. First, I clarify the phenomenon by mentioning threethings that vagueness in our sense (probably) is not.(a) The remark `Someone said something' is naturally described as

vague (who said what?). Similarly, `X is an integer greater than thirty'is an unhelpfully vague hint about the value of X. Vagueness in thissense is underspeci®city, a matter of being less than adequatelyinformative for the purposes in hand. This seems to have nothing todo with borderline cases or with the lack of sharp boundaries: `is aninteger greater than thirty' has sharp boundaries, has no borderlinecases, and is not susceptible to sorites paradoxes. And it is not becauseof any possibility of borderline people or borderline cases of sayingsomething that `someone said something' counts as vague in thealternative sense. I shall ignore the idea of vagueness as underspeci®-city: in philosophical contexts, `vague' has come to be reserved forthe phenomenon I have described.(b) Vagueness must not be straightforwardly identi®ed with para-

digm context-dependence (i.e. having a different extension in differ-ent contexts), even though many terms have both features (e.g. `tall').Fix on a context which can be made as de®nite as you like (inparticular, choose a speci®c comparison class, e.g. current professionalAmerican basketball players): `tall' will remain vague, with borderlinecases and fuzzy boundaries, and the sorites paradox will retain itsforce. This indicates that we are unlikely to understand vagueness orsolve the paradox by concentrating on context-dependence.3

(c) We can also distinguish vagueness from ambiguity. Certainly,terms can be ambiguous and vague: `bank' for example has two quitedifferent main senses (concerning ®nancial institutions or riveredges), both of which are vague. But it is natural to suppose that`tadpole' has a univocal sense, though that sense does not determinea sharp, well-de®ned extension. Certain theories, however, do

ticular, strictly no one is bald unless they have absolutely no hair (see e.g. Sperber andWilson 1986). But even if this line is viable in some cases, it is hopeless for themajority of vague predicates. E.g. should someone count as `tall' only if they are as tallas possible? How about `quite tall'? Or `very hairy'? And where is the strict boundaryof `chair'?

3 There have, however, been some attempts at this type of solution to the soritesparadox using, for example, more elaborate notions of the context of a subject's judge-ment (see e.g. Raffman 1994).


10

attempt to close the gap between vagueness and a form of ambiguity(see chapter 7, §1).

2. type of vague expressions

So far, I have focused on a single dimension of variation associatedwith each vague predicate, such as height for `tall' and number ofgrains for `heap'. But many vague predicates are multi-dimensional:several different dimensions of variation are involved in determiningtheir applicability. The applicability of `big', used to describe people,depends on both height and volume; and even whether somethingcounts as a `heap' depends not only on the number of grains but alsoon their arrangement. And with `nice', for example, there is not evena clear-cut set of dimensions determining the applicability of thepredicate: it is a vague matter which factors are relevant and thedimensions blend into one another.The three central features of vague predicates are shared by multi-

dimensional ones. There are, for example, borderline nice people:indeed, some are borderline because of the multi-dimensionality of`nice', by scoring well in some relevant respects but not in others.Next consider whether multi-dimensional predicates may lack sharpboundaries. In the one-dimensional case, F has a sharp boundary (orsharp boundaries) if possible candidates for it can be ordered with apoint (or points) marking the boundary of F 's extension, so thateverything that falls on one side of the point (or between the points)is F and nothing else is F. For a multi-dimensional predicate, theremay be no uniquely appropriate ordering of possible candidates onwhich to place putative boundary-marking points. (For instance,there is no de®nite ordering of people where each is bigger than theprevious one; in particular, if ordered by height, volume is ignored,and vice versa.) Rather, for a sharply bounded two-dimensionalpredicate the candidates would be more perspicuously set out in atwo-dimensional space in which a boundary could be drawn, wherethe two-dimensional region enclosed by the boundary contains alland only instances of the predicate. With a vague two-dimensionalpredicate no such sharp boundary can be drawn. Similarly, for asharply bounded predicate with a clear-cut set of n dimensions, theboundary would enclose an n-dimensional region containing allof its instances; and vague predicates will lack such a sharp


11

boundary.4 When there is no clear-cut set of dimensions ± for `nice',for example ± this model of boundary-drawing is not so easilyapplied: it is then not possible to construct a suitable arrangement ofcandidates on which to try to draw a boundary of the required sort.But this, I claim, is distinctive of the vagueness of such predicates:they have no sharp boundary, but nor do they have a fuzzy boundaryin the sense of a rough boundary-area of a representative space.`Nice' is so vague that it cannot even be associated with a neat arrayof candidate dimensions, let alone pick out a precise area of such anarray.Finally, multi-dimensional vague predicates are susceptible to

sorites paradoxes. We can construct a sorites series for `heap' byfocusing on the number of grains and minimising the difference inthe arrangement of grains between consecutive members. And for`nice' we could take generosity and consider a series of peoplediffering gradually in this respect, starting with a very mean personand ending with a very generous one, where, for example, otherfeatures relevant to being nice are kept as constant as possible throughthe series.Next, I shall argue that comparatives as well as monadic predicates

can be vague. This has been insuf®ciently recognised and is some-times denied. Cooper 1995, for example, seeks to give an account ofvagueness by explaining how vague monadic predicates depend oncomparatives, taking as a starting point the claim that `classi®ers intheir grammatically positive form [e.g. ``large''] are vague, whilecomparatives are not' (p. 246). With a precise comparative, `F-erthan', for any pair of things x and y, either x is F-er than y, y is F-erthan x, or they are equally F. This will be the case if there is adeterminate ordering of candidates for F-ness (allowing ties). Forexample, there is a one-dimensional ordering of the natural numbersrelating to the comparative `is a smaller number than', and there areno borderline cases of this comparative, which is paradigmaticallyprecise. Since `is a small number' is a vague predicate, this shows howvague positive forms can have precise comparatives. It may seem that

4 Could there be a single, determinate way of balancing the various dimensions of amulti-dimensional predicate that does yield a unique ordering? Perhaps, but this willusually not be the case, and when it is, it may then be appropriate to treat the predicateas one-dimensional, even if the `dimension' is not a natural one. Further discussion ofthis point would need a clearer de®nition of `dimension', but this is not important forour purposes.


12

òlder than' also gives rise to an ordering according to the singledimension of age, and hence that òlder than' must be precise. But, infact, there could be borderline instances of the comparative due toindeterminacy over exactly what should count as the instant ofsomeone's birth and so whether it is before or after the birth ofsomeone else. And such instances illustrate that there is not, in fact, anunproblematic ordering of people for òlder than', even though thereis a total ordering of ages, on which some people cannot be exactlyplaced. Similarly, though there is a single dimension of height, peoplecannot always be exactly placed on it and assigned an exact height.For what exactly should count as the top of one's head? Consequentlythere may also be borderline instances of `taller than'.Comparatives associated with multi-dimensional predicates ± for

example `nicer than' and `more intelligent than' ± are typically vague.They have borderline cases: pairs of people about whom there is nofact of the matter about who is nicer/more intelligent, or whetherthey are equally nice/intelligent. This is particularly common whencomparing people who are nice/intelligent in different ways. Thereare, however, still clear cases of the comparative in addition toborderline cases ± it is not that people are never comparable in respectto niceness ± thus the vague `nicer than', like `nice' itself, has clearpositive, clear negative and borderline cases.Can comparatives also lack sharp boundaries? Talk of boundaries,

whether sharp or fuzzy, is much less natural for comparatives than formonadic predicates. But we might envisage precise comparatives forwhich we could systematically set out ordered pairs of things, hx, yiand draw a sharp boundary around those for which it is true that x isF-er than y. For example, if F has a single dimension then we couldset out pairs in a two-dimensional array, where the x co-ordinate of apair is determined by the location along the dimension of the ®rst ofthe pair, and the y co-ordinate by that of the second. The boundaryline could then be drawn along the diagonal at x = y, where pairsfalling beneath the diagonal are de®nitely true instances of thecomparative `x is F-er than y', and those on or above are de®nitelyfalse. But for many comparatives, including `nicer than', there couldnot be such an arrangement and this gives a sense in which thosecomparatives lack sharp boundaries.Another possible sense in which comparatives may lack sharp

boundaries is the following. Take the comparative `redder than' and


13

choose a purplish-red patch of colour, a. Then consider a series oforangeish-red patches, xi, where xi+1 is redder than xi. It could bede®nitely true that a is redder than x0 (which is nearly orange),de®nitely not true that a is redder than x100, where not only are thereborderline cases of `a is redder than xi' between them, but there is nopoint along the series of xi at which it suddenly stops being the casethat a is redder than xi. So, certain comparatives have borderline casesand exhibit several features akin to the lack of sharp boundaries: theyshould certainly be classi®ed as vague.Having discussed vague monadic predicates and vague compara-

tives, I shall brie¯y mention some other kinds of vague expressions.First, there can be other vague dyadic relational expressions. Forexample, `is a friend of ' has pairs that are borderline cases. Adverbslike `quickly', quanti®ers like `many' and modi®ers like `very' are alsovague. And, just as comparatives can be vague, particularly whenrelated to a multi-dimensional positive, so can superlatives. `Nicest'and `most intelligent' have vague conditions of application: among agroup of people it may be a vague matter, or indeterminate, who isthe nicest or the most intelligent. And vague superlatives provide oneway in which to construct vague singular terms such as `the nicestman' or `the grandest mountain in Scotland', where there is no fact ofthe matter as to which man or mountain the terms pick out. Termswith plural reference like `the high mountains of Scotland' canequally be vague.A theory of vagueness should have the resources to accommodate

all the different types of vague expression. And, for example, weshould reject an account of vagueness that was obliged to deny theabove illustrated features of certain comparatives in order to constructits own account of vague monadic predicates. (See chapter 5, §2about this constraint in connection with degree theories.) The typicalfocus on monadic predicates need not be mistaken, however.Perhaps, as Fine suggests, all vagueness is reducible to predicatevagueness (1975, p. 267), though such a claim needs supportingarguments. Alternatively, vagueness might manifest itself in differentways in different kinds of expression, and this could require takingthose different expression-types in turn and having different criteriaof vagueness for comparatives and monadic predicates. Anotherpossibility is to treat complete sentences as the primary bearers ofvagueness, perhaps in their possession of a non-classical truth-value.


14

This approach would avoid certain tricky questions about whetherthe vagueness of a particular sentence is `due to' a given expression.For example, in a case where it is indeterminate exactly whatmoment a was born and whether it was before the birth of b, wewould avoid the question whether this shows `older than' to bevague, or whether the indeterminacy should be put down to vague-ness in a itself. Provided one can still make sense of a typicalattribution of vagueness to some element of a sentence in theuncontroversial cases, I suggest that this strategy is an appealing one.

3. vagueness in the world?

Is it only linguistic items ± words or phrases ± that can be vague?Surely not: thoughts and beliefs are among the mental items whichshare the central characteristics of vagueness; other controversial casesinclude perceptions. What about the world itself: could the world bevague as well as our descriptions of it? Can there be vague objects?Or vague properties (the ontic correlates of predicates)? Consider BenNevis: any sharp spatio-temporal boundaries drawn around themountain would be arbitrarily placed, and would not re¯ect a naturalboundary. So it may seem that Ben Nevis has fuzzy boundaries, andso, given the common view that a vague object is an object withfuzzy, spatio-temporal boundaries, that it is a vague object. (See e.g.Parsons 1987, Tye 1990 and Zemach 1991 for arguments that thereare vague objects.) But there are, of course, other contendingdescriptions of the situation here. For example, perhaps the onlyobjects we should admit into our ontology are precise/sharp althoughwe fail to pick out a single one of them with our (vague) name `BenNevis'. It would then be at the level of our representations of theworld that vagueness came in. (See chapter 7, §1 on an indeterminatereference view.)5

My concern is with linguistic vagueness and I shall generally ignoreontic vagueness. This would be a mistake if a theory of linguisticvagueness had to rely on ontic vagueness. But that would besurprising since it seems at least possible to have vague language in anon-vague world. In particular, even if all objects, properties and

5 The most discussed strand of the ontic vagueness debate focuses on Evans's formalargument which aims to establish a negative answer to his question `Can there bevague objects?' (1978; see Keefe and Smith 1997b, §5 for an overview of the debate).


15

facts were precise, we would still have reason, for everyday purposes,to use a vague expression such as `tall', which would still haveborderline cases (even if those cases could also be described in non-vague terms involving precise heights etc.). Similarly, in a preciseworld we would still use vague singular terms, perhaps to pick outvarious large collections of precise fundamental particulars (e.g. asclouds or mountains) where the boundaries of those collections areleft fuzzy. So it seems that language could still be vague if the concreteworld were precise.6

The theories of vagueness of this book are theories of linguisticvagueness and in the next section I brie¯y introduce them.

4. theories of vagueness

The candidate theories of vagueness can be systematically surveyed byconsidering how they address two central tasks. The ®rst is to identifythe logic and semantics for a vague language ± a task bound up withproviding an account of borderline cases and of fuzzy boundaries.The second task is that of addressing the sorites paradox.

(i) The logic and semantics of vagueness

The simplest approach is to retain classical logic and semantics.Borderline case predications are either true or false after all, thoughwe do not and cannot know which. Similarly, despite appearances,vague predicates have well-de®ned extensions: there is a sharpboundary between the tall people and the rest, and between the redshades of the spectrum and the other colours. As chapter 3 willdescribe, the epistemic view takes this line and accounts for vaguenessin terms of our ignorance ± for example, ignorance of where thesharp boundaries to our vague predicates lie. And a pragmatic account ofvagueness also seeks to avoid challenging classical logic and semantics,but this time by accounting for vagueness in terms of pragmaticrelations between speakers and their language: see chapter 6.

6 These are only prima facie reasons for not approaching linguistic vagueness via onticvagueness: a tighter case would require clari®cation of what vagueness in the worldwould be. They also do not seem to bear on the question whether there can be vaguesets, which might also be counted as a form of ontic vagueness. Tye, for example,believes that there are vague sets and maintains that they are crucial to his own theoryof the linguistic phenomena (see Tye 1990).


16

If we do not retain classical logic and semantics, we can say insteadthat when a is a borderline case of F, the truth-value of à is F ' is, asMachina puts it, ìn some way peculiar, or indeterminate or lackingentirely' (1976, p. 48). This generates a number of non-classicaloptions.Note that a borderline case of the predicate F is equally a border-

line case of not-F: it is unclear whether or not the candidate is F. Thissymmetry prevents us from simply counting a borderline F as not-F.But there are several ways of respecting this symmetry. Some take theline that a predication in a borderline case is both true and false: thereis a truth-value glut. This can be formalised within the context of aparaconsistent logic ± a logic that admits true contradictions (seeHyde 1997 and chapter 7, §7 for discussion of that view).A more popular position is to admit truth-value gaps: borderline

predications are neither true nor false. One elegant development issupervaluationism. The basic idea is that a proposition involving thevague predicate `tall', for example, is true (false) if it comes out true(false) on all the ways in which we can make `tall' precise (ways, thatis, which preserve the truth-values of uncontentiously true or falsecases of à is tall'). A borderline case, `Tek is tall', will be neither truenor false, for it is true on some ways of making `tall' precise and falseon others. But a classical tautology like èither Tek is tall or he is nottall' will still come out true because wherever a sharp boundary for`tall' is drawn, that compound sentence will come out true. In thisway, the supervaluationist adopts a non-classical semantics whileaiming to minimise divergence from classical logic. A theory of thistype will be defended in chapters 7 and 8.Rather than holding that predications in borderline cases lack a

truth-value, another option is to hold that they have a third value ±`neutral', ìndeterminate' or ìnde®nite' ± leading to a three-valuedlogic (see chapter 4). Alternatively, degree theories countenance degreesof truth, introducing a whole spectrum of truth-values from 0 to 1,with complete falsity as degree 0 and complete truth as degree 1.Borderline cases each take some value between 0 and 1, with `x isred' gradually increasing in truth-value as we move along the colourspectrum from orange to red. This calls for an in®nite-valued logic ora so-called `fuzzy logic', and there have been a variety of differentversions (see chapter 4).So far the sketched positions at least agree that there is some positive


17

account to be given of the logic and semantics of vagueness. Otherwriters have taken a more pessimistic line. In particular, Russellclaims that logic assumes precision, and since natural language is notprecise it cannot be in the province of logic at all (1923, pp. 88±9). Ifsuch a `no logic' thesis requires wholesale rejection of reasoning withvague predicates ± and hence of most reasoning in natural language ±it is absurdly extreme. And arguments involving vague predicates areclearly not all on a par. For example, `anyone with less than 500 hairson his head is bald; Fred has less than 500 hairs on his head; thereforeFred is bald' is an unproblematically good argument (from Cargile1969, pp. 196±7). And, similarly, there are other ways of arguingwith vague predicates that should certainly be rejected. Some accountis needed of inferences that are acceptable and others that fail, and tosearch for systematic principles capturing this is to seek elements of alogic of vague language. So, I take the pessimism of the no-logicapproach to be a very last resort, and in this book I concentrate onmore positive approaches.Focusing on the question how borderline case predication should

be classi®ed, we seem to have exhausted the possibilities. They maybe true or false, or have no truth value at all (in particular, beingneither true nor false), or be both true and false, or have a non-classical value from some range of values. When it comes to surveyingsolutions to the sorites paradox, however, there may additionally bealternatives that do not provide a theory of vagueness and perhaps donot answer the question how borderline cases are to be classi®ed. Iconcentrate on those which do ®t into a theory of vagueness.

(ii) The sorites paradox

A paradigm sorites set-up for the predicate F is a sequence of objectsxi, such that the two premises

(1) Fx1(2) For all i, if Fxi then Fxi+1

both appear true, but, for some suitably large n, the putative conclusion

(3) Fxn

seems false. For example, in the case of `tall', the xi might be theseries of men described earlier, each a hundredth of an inch shorter


18

than the previous one and where x1 is seven feet tall. (1) `x1 is tall' isthen true; and so, it seems, is the inductive premise, (2) `for all i, if xiis tall, so is xi+1'. But it is surely false that (3) x3000 ± who is only 4feet 6 inches ± is tall.A second form of sorites paradox can be constructed when, instead

of the quanti®ed inductive premise (2), we start with a collection ofparticular conditional premises, (2Ci), each of the form `if Fxi thenFxi+1'. For example,

(2C1) if x1 is tall, so is x2(2C2) if x2 is tall, so is x3

and so on. And the use of conditionals is not essential: we can take asequence of premises of the form : (Fxi & :Fxi+1) ± a formulationthat goes back at least to Diogenes Laertius (see Long and Sedley1987, p. 222). Alternatively, (2) could be replaced by a quanti®cationover the negated conjunctions of that form.As well as needing to solve the paradox, we must assess that general

form of argument because it is used both in philosophical argumentsoutside the discussion of vagueness (e.g. with the story of the ship ofTheseus) and in various more everyday debates (the slippery slopearguments mentioned in §1).7

Responses to a sorites paradox can be divided into four types. Wecan:(a) deny the validity of the argument, refusing to grant that the

conclusion follows from the given premises; or(b) question the strict truth of the general inductive premise (2) or

of at least one of the conditionals (2Ci); or(c) accept the validity of the argument and the truth of its inductive

premise (or of all the conditional premises) but contest the supposedtruth of premise (1) or the supposed falsity of the conclusion (3); or(d) grant that there are compelling reasons both to take the

7 As a further example of the former, consider Kirk 1986 (pp. 217ff ). RegardingQuine's thesis about the indeterminacy of translation, Kirk uses an argument with theform of the quanti®cational version of the paradox to argue that there can be no inde-terminacy of translation because, ®rst, there would be no indeterminacy in translatingbetween the languages of infants each of whom is at an early stage of language-acquisi-tion and, second, if there is no indeterminacy at one step of acquisition then there isnone at the next. He presents his argument as using mathematical induction but doesnot ask whether its employment of vague predicates casts doubt on that mode of argu-ment.


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argument form as valid, and to accept the premises and deny theconclusion, concluding that this demonstrates the incoherence of thepredicate in question.

I shall brie¯y survey these in turn, ignoring here the questionwhether we should expect a uniform solution to all sorites paradoxeswhatever their form and whatever predicate is involved. (Wright1987 argues that different responses could be required depending onthe reasons that support the inductive premise.) Any response mustexplain away apparent dif®culties with accepting the selected solu-tion; for example, if the main premise is denied, it must be explainedwhy that premise is so plausible. More generally, a theory shouldaccount for the persuasiveness of the paradox as a paradox and shouldexplain how this is compatible with the fact that we are never, orvery rarely, actually led into contradiction.(a) Denying the validity of the sorites argument seems to require

giving up absolutely fundamental rules of inference. This can be seenmost clearly when the argument takes the second form involving aseries of conditionals, the (2Ci). The only rule of inference neededfor this argument is modus ponens. Dummett argues that this rulecannot be given up, as it is constitutive of the meaning of `if ' thatmodus ponens is valid (1975, p. 306). To derive the conclusion inthe ®rst form of sorites, we only need universal instantiation inaddition to modus ponens; but, as Dummett again argues, universalinstantiation seems too central to the meaning of `all' to be reasonablychallenged (1975, p. 306). I agree on both points and shall not pursuethe matter further here.There is, however, a different way of rejecting the validity of the

many-conditionals form of the sorites. It might be suggested thateven though each step is acceptable on its own, chaining too manysteps does not guarantee the preservation of truth if what counts aspreserving truth is itself a vague matter. (And then the ®rst form ofsorites could perhaps be rejected on the grounds that it is in effectshort hand for a multi-conditional argument.) As Dummett againnotes, this is to deny the transitivity of validity, which would beanother drastic move, given that chaining inferences is normallytaken to be essential to the very enterprise of proof.8

8 But see Parikh 1983. In my chapter 4, §7 the possibility is brie¯y entertained.


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Rather than questioning particular inference rules or the ways theycan be combined, Russell's global rejection of logic for vague naturallanguage leads him to dismiss `the old puzzle about the man whowent bald', simply on the grounds that `bald' is vague (1923, p. 85).The sorites arguments, on his view, cannot be valid because, con-taining vague expressions, they are just not the kind of thing that canbe valid or invalid.(b) If we take a formulation of the paradox that uses negated

conjunctions (or assume that ìf ' is captured by the material condi-tional), then within a classical framework denying the quanti®edinductive premise or one of its instances commits us to there being ani such that `Fxi and not-Fxi+1' is true. This implies the existence ofsharp boundaries and the epistemic theorist, who takes this line, willexplain why vague predicates appear not to draw sharp boundaries byreference to our ignorance (see chapter 3).In a non-classical framework there is a wide variety of ways of

developing option (b), and it is not clear or uncontroversial which ofthese entail a commitment to sharp boundaries. For example, thesupervaluationist holds that the generalised premise (2) `for all i, if Fxithen Fxi+1' is false: for each F* which constitutes a way of making Fprecise, there will be some xi or other which is the last F* and isfollowed by an xi+1 which is not-F*. But since there is no particular ifor which `Fxi and not-Fxi+1' is true ± i.e. true however F is madeprecise ± supervaluationists claim that their denial of (2) does notmean accepting that F is sharply bounded (see chapter 7). And othernon-classical frameworks may allow that (2) is not true, while notaccepting that it is false. Tye 1994, for example, maintains that theinductive premise and its negation both take his intermediate truth-value, ìnde®nite'.Degree theorists offer another non-classical version of option (b):

they can deny that the premises are strictly true while maintaining thatthey are nearly true. The essence of their account is to hold that thepredications Fxi take degrees of truth that encompass a graduallydecreasing series from complete truth (degree 1) to complete falsity(degree 0). There is never a substantial drop in degree of truth betweenconsecutive Fxi; so, given a natural interpretation of the conditional,the particular premises ìf Fxi then Fxi+1' can each come out at leastvery nearly true, though some are not completely true. If the soritesargument based on many conditionals is to count as strictly valid, then


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an account of validity is needed that allows a valid argument to havenearly true premises but a false conclusion. But with some degree-theoretic accounts of validity, the sorites fails to be valid ± thus a degreetheorist can combine responses (a) and (b) (see chapter 4, §7).Intuitionistic logic opens up the possibility of another non-classical

position that can respond to the sorites by denying the inductivepremise (2), while not accepting the classical equivalent of this denial,(9xi)(Fxi & :Fxi+1), which is the unwanted assertion of sharpboundaries. Putnam 1983 suggests this strategy. But critics haveshown that with various reasonable additional assumptions, otherversions of sorites arguments still lead to paradox. In particular, if, asmight be expected, you adopt intuitionistic semantics as well asintuitionistic logic, paradoxes recur (see Read and Wright 1985).And Williamson 1996 shows that combining Putnam's approach tovagueness with his epistemological conception of truth still facesparadox. (See also Chambers 1998, who argues that, given Putnam'sown view on what would make for vagueness, paradox againemerges.) The bulk of the criticisms point to the conclusion thatthere is no sustainable account of vagueness that emerges fromrejecting classical logic in favour of intuitionistic logic.(c) Take the sorites (H+) with the premises òne grain of sand is

not a heap' and àdding a single grain to a non-heap will not turn itinto a heap'. If we accept these premises and the validity of theargument, it follows that we will never get a heap, no matter howmany grains are piled up: so there are no heaps. Similarly, soritesparadoxes for `bald', `tall' and `person' could be taken to show thatthere are no bald people, no tall people and indeed no people at all.Unger bites the bullet and takes this nihilistic line, summarised in thetitle of one of his papers: `There are no ordinary things' (Unger 1979;see also Wheeler 1975, 1979 and Heller 1988).The thesis, put in linguistic terms, is that all vague predicates lack

serious application, i.e. they apply either to nothing (ìs a heap') or toeverything (ìs not a heap'). Classical logic can be retained in itsentirety, but sharp boundaries are avoided by denying that vaguepredicates succeed in drawing any boundaries, fuzzy or otherwise.There will be no borderline cases: for any vague F, everything is F oreverything is not-F, and thus nothing is borderline F.9

9 See Williamson 1994, chapter 6, for a sustained attack on various forms of nihilism.For example, he shows how the nihilist cannot state or argue for his own position on


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The response of accepting the conclusion of every sorites paradoxcannot be consistently sustained. For in addition to (H+), there is theargument (H7) with the premises `ten thousand grains make a heap'and `removing one grain from a heap still leaves a heap', leading tothe conclusion that a single grain of sand is a heap, which isincompatible with the conclusion of (H+). Such reversibility istypical; given a sorites series of items, the argument can be run eitherway through them. Unger's response to (H7) would be to deny theinitial premise: there are no heaps ± as (H+) supposedly shows us ± soit is not true that ten thousand grains make a heap. Systematicgrounds would then be needed to enable us to decide which of a pairof sorites paradoxes is sound (e.g. why there are no heaps rather thaneverything being a heap).Unger is driven to such an extreme position by the strength of the

arguments in support of the inductive premises of sorites paradoxes. Ifour words determined sharp boundaries, Unger claims, our under-standing of them would be a miracle of conceptual comprehension(1979, p. 126). The inductive premise, guaranteeing this lack of sharpboundaries, re¯ects a semantic rule central to the meaning of thevague F. But, we should ask Unger, can the tolerance principleexpressed in the inductive premise for `tall' really be more certainthan the truth of the simple predication of `tall' to a seven-foot man?Is it plausible to suppose that the expression `tall' is meaningful andconsistent but that there could not be anything tall, when learningthe term typically involves ostension and hence confrontation withalleged examples? A different miracle of conceptual comprehensionwould be needed then to explain how we can understand thatmeaning and, in general, how we can use such empty predicatessuccessfully to communicate anything at all. It may be more plausibleto suppose that if there are any rules governing the application of`tall', then, in addition to tolerance rules, there are ones dictating that`tall' applies to various paradigmatic cases and does not apply tovarious paradigmatically short people. Sorites paradoxes could thendemonstrate the inconsistency of such a set of rules, and this is option(d).Responses (c) and (d) are not always clearly distinguished. Writers

his own terms (e.g. the expressions he tries to use must count as incoherent). My dis-cussion of methodological matters in chapter 2 will suggest that a swifter rejection ofthe position is warranted anyway.


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like Unger are primarily concerned with drawing ontological conclu-sions. It is enough for them to emphasise the tolerance of a predicatelike `tall' which already guarantees, they claim, that the worldcontains nothing that strictly answers to that description: they are notso concerned to examine what further rules might govern thepredicate and perhaps render it incoherent. But other writers, forexample Dummett, explore these conceptual questions.(d) Having argued in detail against alternative responses to the

paradox, Dummett 1975 maintains that there is no choice but toaccept that a sorites paradox for F exempli®es an undeniably validform of argument from what the semantic rules for F dictate to betrue premises to what they dictate to be a false conclusion. Theparadoxes thus reveal the incoherence of the rules governing vagueterms: by simply following those rules, speakers could be led tocontradict themselves. This inconsistency means that there can be nocoherent logic governing vague language.10

Once (d)-theorists have concluded that vague predicates are in-coherent, they may agree with Russell that such predicates cannotappear in valid arguments. So option (d) can be developed in such away that makes it compatible with option (a), though this route tothe denial of validity is very different from Russell's. (Being outsidethe scope of logic need not make for incoherence.)The acceptance of such pervasive inconsistency is highly undesir-

able and such pessimism is premature; and it is even by Dummett'sown lights a pessimistic response to the paradox, adopted as a lastresort rather than as a positive treatment of the paradox that stands ascompetitor to any other promising alternatives. Communicationusing vague language is overwhelmingly successful and we are neverin practice driven to incoherence (a point stressed by Wright, e.g.1987, p. 236). And even when shown the sorites paradox, we arerarely inclined to revise our initial judgement of the last member ofthe series. It looks unlikely that the success and coherence in ourpractice is owed to our grasp of inconsistent rules. A defence of someversion of option (a) or (b) would provide an attractive way of

10 See also Rolf 1981, 1984. Horgan 1994, 1998 advocates a different type of theinconsistency view. He agrees that sorites paradoxes (and other related arguments)demonstrate logical incoherence, but considers that incoherence to be tempered orinsulated, so that it does not infect the whole language and allows us to use thelanguage successfully despite the incoherence (see chapter 8, §2).


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rosanna keefe - library of congress

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